Investigation of Graphs Having Identical Path Layer Matrices

The path layer matrix of a graph G, discussed by Dobrynin [1, 2, 3] and Yang et al. [9], is the matrix τ(G) = ‖τi,j‖, where i = 1, 2, . . . , p(G) and j = 1, 2, . . . , p(G) − 1, and the value of τi,j is the number of paths with initial vertex vi having length j [1]. In this paper we seek to determine the minimal order p such that there exist nonisomorphic graphs G1 and G2 with order p and τ(G1) = τ(G2). The problem is posed for graphs in general, as well as for specialized cases of graphs, including r-regular graphs, in which each vertex is connected to r other vertices. We construct a pair of nonisomorphic 5-regular graphs G20−1 and H20−1 of order 20 such that τ(G20−1) = τ(H20−1). We approach various aspects of the problem in several interesting manners and furthermore attempt to generalize the results obtained in [8] for graphs of larger regularities.